Abstract: Let be any finitely generated group, and let be a field of characteristic . It is shown that the graded group ring satisfies a nontrivial polynomial identity if and only if the pro- completion of is -adic analytic, i.e. can be given the structure of a Lie group over the -adic field .

The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.